# Properties

 Label 2025.2.b.g.649.3 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 405) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.3 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.g.649.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.732051i q^{2} +1.46410 q^{4} -4.73205i q^{7} +2.53590i q^{8} +O(q^{10})$$ $$q+0.732051i q^{2} +1.46410 q^{4} -4.73205i q^{7} +2.53590i q^{8} -5.73205 q^{11} -1.46410i q^{13} +3.46410 q^{14} +1.07180 q^{16} -2.73205i q^{17} -4.46410 q^{19} -4.19615i q^{22} +3.46410i q^{23} +1.07180 q^{26} -6.92820i q^{28} -3.19615 q^{29} -3.00000 q^{31} +5.85641i q^{32} +2.00000 q^{34} -2.73205i q^{37} -3.26795i q^{38} -7.19615 q^{41} -0.196152i q^{43} -8.39230 q^{44} -2.53590 q^{46} -8.73205i q^{47} -15.3923 q^{49} -2.14359i q^{52} -6.73205i q^{53} +12.0000 q^{56} -2.33975i q^{58} +8.26795 q^{59} +4.00000 q^{61} -2.19615i q^{62} -2.14359 q^{64} +3.46410i q^{67} -4.00000i q^{68} -3.73205 q^{71} +7.66025i q^{73} +2.00000 q^{74} -6.53590 q^{76} +27.1244i q^{77} -15.4641 q^{79} -5.26795i q^{82} -2.19615i q^{83} +0.143594 q^{86} -14.5359i q^{88} -5.19615 q^{89} -6.92820 q^{91} +5.07180i q^{92} +6.39230 q^{94} -9.66025i q^{97} -11.2679i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4}+O(q^{10})$$ 4 * q - 8 * q^4 $$4 q - 8 q^{4} - 16 q^{11} + 32 q^{16} - 4 q^{19} + 32 q^{26} + 8 q^{29} - 12 q^{31} + 8 q^{34} - 8 q^{41} + 8 q^{44} - 24 q^{46} - 20 q^{49} + 48 q^{56} + 40 q^{59} + 16 q^{61} - 64 q^{64} - 8 q^{71} + 8 q^{74} - 40 q^{76} - 48 q^{79} + 56 q^{86} - 16 q^{94}+O(q^{100})$$ 4 * q - 8 * q^4 - 16 * q^11 + 32 * q^16 - 4 * q^19 + 32 * q^26 + 8 * q^29 - 12 * q^31 + 8 * q^34 - 8 * q^41 + 8 * q^44 - 24 * q^46 - 20 * q^49 + 48 * q^56 + 40 * q^59 + 16 * q^61 - 64 * q^64 - 8 * q^71 + 8 * q^74 - 40 * q^76 - 48 * q^79 + 56 * q^86 - 16 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.732051i 0.517638i 0.965926 + 0.258819i $$0.0833333\pi$$
−0.965926 + 0.258819i $$0.916667\pi$$
$$3$$ 0 0
$$4$$ 1.46410 0.732051
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.73205i − 1.78855i −0.447521 0.894274i $$-0.647693\pi$$
0.447521 0.894274i $$-0.352307\pi$$
$$8$$ 2.53590i 0.896575i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.73205 −1.72828 −0.864139 0.503253i $$-0.832136\pi$$
−0.864139 + 0.503253i $$0.832136\pi$$
$$12$$ 0 0
$$13$$ − 1.46410i − 0.406069i −0.979172 0.203034i $$-0.934920\pi$$
0.979172 0.203034i $$-0.0650803\pi$$
$$14$$ 3.46410 0.925820
$$15$$ 0 0
$$16$$ 1.07180 0.267949
$$17$$ − 2.73205i − 0.662620i −0.943522 0.331310i $$-0.892509\pi$$
0.943522 0.331310i $$-0.107491\pi$$
$$18$$ 0 0
$$19$$ −4.46410 −1.02414 −0.512068 0.858945i $$-0.671120\pi$$
−0.512068 + 0.858945i $$0.671120\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.19615i − 0.894623i
$$23$$ 3.46410i 0.722315i 0.932505 + 0.361158i $$0.117618\pi$$
−0.932505 + 0.361158i $$0.882382\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.07180 0.210197
$$27$$ 0 0
$$28$$ − 6.92820i − 1.30931i
$$29$$ −3.19615 −0.593511 −0.296755 0.954954i $$-0.595905\pi$$
−0.296755 + 0.954954i $$0.595905\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 5.85641i 1.03528i
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.73205i − 0.449146i −0.974457 0.224573i $$-0.927901\pi$$
0.974457 0.224573i $$-0.0720988\pi$$
$$38$$ − 3.26795i − 0.530131i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.19615 −1.12385 −0.561925 0.827188i $$-0.689939\pi$$
−0.561925 + 0.827188i $$0.689939\pi$$
$$42$$ 0 0
$$43$$ − 0.196152i − 0.0299130i −0.999888 0.0149565i $$-0.995239\pi$$
0.999888 0.0149565i $$-0.00476097\pi$$
$$44$$ −8.39230 −1.26519
$$45$$ 0 0
$$46$$ −2.53590 −0.373898
$$47$$ − 8.73205i − 1.27370i −0.770988 0.636850i $$-0.780237\pi$$
0.770988 0.636850i $$-0.219763\pi$$
$$48$$ 0 0
$$49$$ −15.3923 −2.19890
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.14359i − 0.297263i
$$53$$ − 6.73205i − 0.924718i −0.886693 0.462359i $$-0.847003\pi$$
0.886693 0.462359i $$-0.152997\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 12.0000 1.60357
$$57$$ 0 0
$$58$$ − 2.33975i − 0.307224i
$$59$$ 8.26795 1.07640 0.538198 0.842819i $$-0.319105\pi$$
0.538198 + 0.842819i $$0.319105\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ − 2.19615i − 0.278912i
$$63$$ 0 0
$$64$$ −2.14359 −0.267949
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.46410i 0.423207i 0.977356 + 0.211604i $$0.0678686\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ − 4.00000i − 0.485071i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.73205 −0.442913 −0.221456 0.975170i $$-0.571081\pi$$
−0.221456 + 0.975170i $$0.571081\pi$$
$$72$$ 0 0
$$73$$ 7.66025i 0.896565i 0.893892 + 0.448282i $$0.147964\pi$$
−0.893892 + 0.448282i $$0.852036\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −6.53590 −0.749719
$$77$$ 27.1244i 3.09111i
$$78$$ 0 0
$$79$$ −15.4641 −1.73985 −0.869924 0.493186i $$-0.835832\pi$$
−0.869924 + 0.493186i $$0.835832\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 5.26795i − 0.581748i
$$83$$ − 2.19615i − 0.241059i −0.992710 0.120530i $$-0.961541\pi$$
0.992710 0.120530i $$-0.0384592\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0.143594 0.0154841
$$87$$ 0 0
$$88$$ − 14.5359i − 1.54953i
$$89$$ −5.19615 −0.550791 −0.275396 0.961331i $$-0.588809\pi$$
−0.275396 + 0.961331i $$0.588809\pi$$
$$90$$ 0 0
$$91$$ −6.92820 −0.726273
$$92$$ 5.07180i 0.528771i
$$93$$ 0 0
$$94$$ 6.39230 0.659316
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 9.66025i − 0.980850i −0.871483 0.490425i $$-0.836842\pi$$
0.871483 0.490425i $$-0.163158\pi$$
$$98$$ − 11.2679i − 1.13823i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −2.66025 −0.264705 −0.132353 0.991203i $$-0.542253\pi$$
−0.132353 + 0.991203i $$0.542253\pi$$
$$102$$ 0 0
$$103$$ 0.535898i 0.0528036i 0.999651 + 0.0264018i $$0.00840494\pi$$
−0.999651 + 0.0264018i $$0.991595\pi$$
$$104$$ 3.71281 0.364071
$$105$$ 0 0
$$106$$ 4.92820 0.478669
$$107$$ − 8.53590i − 0.825196i −0.910913 0.412598i $$-0.864621\pi$$
0.910913 0.412598i $$-0.135379\pi$$
$$108$$ 0 0
$$109$$ 6.07180 0.581573 0.290786 0.956788i $$-0.406083\pi$$
0.290786 + 0.956788i $$0.406083\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 5.07180i − 0.479240i
$$113$$ 19.1244i 1.79907i 0.436851 + 0.899534i $$0.356094\pi$$
−0.436851 + 0.899534i $$0.643906\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.67949 −0.434480
$$117$$ 0 0
$$118$$ 6.05256i 0.557183i
$$119$$ −12.9282 −1.18513
$$120$$ 0 0
$$121$$ 21.8564 1.98695
$$122$$ 2.92820i 0.265107i
$$123$$ 0 0
$$124$$ −4.39230 −0.394441
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 14.5885i − 1.29452i −0.762271 0.647258i $$-0.775916\pi$$
0.762271 0.647258i $$-0.224084\pi$$
$$128$$ 10.1436i 0.896575i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.5885 1.36197 0.680985 0.732297i $$-0.261552\pi$$
0.680985 + 0.732297i $$0.261552\pi$$
$$132$$ 0 0
$$133$$ 21.1244i 1.83171i
$$134$$ −2.53590 −0.219068
$$135$$ 0 0
$$136$$ 6.92820 0.594089
$$137$$ 2.53590i 0.216656i 0.994115 + 0.108328i $$0.0345497\pi$$
−0.994115 + 0.108328i $$0.965450\pi$$
$$138$$ 0 0
$$139$$ −0.607695 −0.0515440 −0.0257720 0.999668i $$-0.508204\pi$$
−0.0257720 + 0.999668i $$0.508204\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 2.73205i − 0.229269i
$$143$$ 8.39230i 0.701800i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −5.60770 −0.464096
$$147$$ 0 0
$$148$$ − 4.00000i − 0.328798i
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ 5.39230 0.438820 0.219410 0.975633i $$-0.429587\pi$$
0.219410 + 0.975633i $$0.429587\pi$$
$$152$$ − 11.3205i − 0.918214i
$$153$$ 0 0
$$154$$ −19.8564 −1.60007
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 19.1244i − 1.52629i −0.646227 0.763145i $$-0.723654\pi$$
0.646227 0.763145i $$-0.276346\pi$$
$$158$$ − 11.3205i − 0.900611i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.3923 1.29189
$$162$$ 0 0
$$163$$ − 12.7321i − 0.997251i −0.866817 0.498626i $$-0.833838\pi$$
0.866817 0.498626i $$-0.166162\pi$$
$$164$$ −10.5359 −0.822715
$$165$$ 0 0
$$166$$ 1.60770 0.124781
$$167$$ − 17.6603i − 1.36659i −0.730142 0.683296i $$-0.760546\pi$$
0.730142 0.683296i $$-0.239454\pi$$
$$168$$ 0 0
$$169$$ 10.8564 0.835108
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 0.287187i − 0.0218978i
$$173$$ − 8.53590i − 0.648972i −0.945890 0.324486i $$-0.894809\pi$$
0.945890 0.324486i $$-0.105191\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −6.14359 −0.463091
$$177$$ 0 0
$$178$$ − 3.80385i − 0.285110i
$$179$$ −8.12436 −0.607243 −0.303621 0.952793i $$-0.598196\pi$$
−0.303621 + 0.952793i $$0.598196\pi$$
$$180$$ 0 0
$$181$$ 26.4641 1.96706 0.983531 0.180742i $$-0.0578498\pi$$
0.983531 + 0.180742i $$0.0578498\pi$$
$$182$$ − 5.07180i − 0.375947i
$$183$$ 0 0
$$184$$ −8.78461 −0.647610
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 15.6603i 1.14519i
$$188$$ − 12.7846i − 0.932413i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.12436 −0.587858 −0.293929 0.955827i $$-0.594963\pi$$
−0.293929 + 0.955827i $$0.594963\pi$$
$$192$$ 0 0
$$193$$ 5.26795i 0.379195i 0.981862 + 0.189598i $$0.0607184\pi$$
−0.981862 + 0.189598i $$0.939282\pi$$
$$194$$ 7.07180 0.507725
$$195$$ 0 0
$$196$$ −22.5359 −1.60971
$$197$$ − 13.8564i − 0.987228i −0.869681 0.493614i $$-0.835676\pi$$
0.869681 0.493614i $$-0.164324\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 1.94744i − 0.137021i
$$203$$ 15.1244i 1.06152i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −0.392305 −0.0273332
$$207$$ 0 0
$$208$$ − 1.56922i − 0.108806i
$$209$$ 25.5885 1.76999
$$210$$ 0 0
$$211$$ −8.85641 −0.609700 −0.304850 0.952400i $$-0.598606\pi$$
−0.304850 + 0.952400i $$0.598606\pi$$
$$212$$ − 9.85641i − 0.676941i
$$213$$ 0 0
$$214$$ 6.24871 0.427153
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 14.1962i 0.963698i
$$218$$ 4.44486i 0.301044i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ − 16.7846i − 1.12398i −0.827144 0.561990i $$-0.810036\pi$$
0.827144 0.561990i $$-0.189964\pi$$
$$224$$ 27.7128 1.85164
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 18.0526i − 1.19819i −0.800678 0.599095i $$-0.795527\pi$$
0.800678 0.599095i $$-0.204473\pi$$
$$228$$ 0 0
$$229$$ 12.0000 0.792982 0.396491 0.918039i $$-0.370228\pi$$
0.396491 + 0.918039i $$0.370228\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 8.10512i − 0.532127i
$$233$$ 28.0526i 1.83778i 0.394509 + 0.918892i $$0.370915\pi$$
−0.394509 + 0.918892i $$0.629085\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.1051 0.787976
$$237$$ 0 0
$$238$$ − 9.46410i − 0.613467i
$$239$$ 0.535898 0.0346644 0.0173322 0.999850i $$-0.494483\pi$$
0.0173322 + 0.999850i $$0.494483\pi$$
$$240$$ 0 0
$$241$$ −16.3205 −1.05130 −0.525648 0.850702i $$-0.676177\pi$$
−0.525648 + 0.850702i $$0.676177\pi$$
$$242$$ 16.0000i 1.02852i
$$243$$ 0 0
$$244$$ 5.85641 0.374918
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.53590i 0.415869i
$$248$$ − 7.60770i − 0.483089i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −10.3923 −0.655956 −0.327978 0.944685i $$-0.606367\pi$$
−0.327978 + 0.944685i $$0.606367\pi$$
$$252$$ 0 0
$$253$$ − 19.8564i − 1.24836i
$$254$$ 10.6795 0.670091
$$255$$ 0 0
$$256$$ −11.7128 −0.732051
$$257$$ 10.3923i 0.648254i 0.946014 + 0.324127i $$0.105071\pi$$
−0.946014 + 0.324127i $$0.894929\pi$$
$$258$$ 0 0
$$259$$ −12.9282 −0.803319
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 11.4115i 0.705007i
$$263$$ − 21.3205i − 1.31468i −0.753595 0.657339i $$-0.771682\pi$$
0.753595 0.657339i $$-0.228318\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −15.4641 −0.948165
$$267$$ 0 0
$$268$$ 5.07180i 0.309809i
$$269$$ −10.6603 −0.649967 −0.324984 0.945720i $$-0.605359\pi$$
−0.324984 + 0.945720i $$0.605359\pi$$
$$270$$ 0 0
$$271$$ −2.92820 −0.177876 −0.0889378 0.996037i $$-0.528347\pi$$
−0.0889378 + 0.996037i $$0.528347\pi$$
$$272$$ − 2.92820i − 0.177548i
$$273$$ 0 0
$$274$$ −1.85641 −0.112150
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3.80385i 0.228551i 0.993449 + 0.114276i $$0.0364547\pi$$
−0.993449 + 0.114276i $$0.963545\pi$$
$$278$$ − 0.444864i − 0.0266812i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.4641 0.922511 0.461255 0.887267i $$-0.347399\pi$$
0.461255 + 0.887267i $$0.347399\pi$$
$$282$$ 0 0
$$283$$ 29.3205i 1.74292i 0.490464 + 0.871462i $$0.336827\pi$$
−0.490464 + 0.871462i $$0.663173\pi$$
$$284$$ −5.46410 −0.324235
$$285$$ 0 0
$$286$$ −6.14359 −0.363278
$$287$$ 34.0526i 2.01006i
$$288$$ 0 0
$$289$$ 9.53590 0.560935
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.2154i 0.656331i
$$293$$ − 28.7321i − 1.67854i −0.543712 0.839272i $$-0.682981\pi$$
0.543712 0.839272i $$-0.317019\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.92820 0.402694
$$297$$ 0 0
$$298$$ 5.85641i 0.339253i
$$299$$ 5.07180 0.293310
$$300$$ 0 0
$$301$$ −0.928203 −0.0535007
$$302$$ 3.94744i 0.227150i
$$303$$ 0 0
$$304$$ −4.78461 −0.274416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 14.0526i 0.802022i 0.916073 + 0.401011i $$0.131341\pi$$
−0.916073 + 0.401011i $$0.868659\pi$$
$$308$$ 39.7128i 2.26285i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 19.7321 1.11890 0.559451 0.828863i $$-0.311012\pi$$
0.559451 + 0.828863i $$0.311012\pi$$
$$312$$ 0 0
$$313$$ 9.07180i 0.512768i 0.966575 + 0.256384i $$0.0825312\pi$$
−0.966575 + 0.256384i $$0.917469\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ −22.6410 −1.27366
$$317$$ 6.19615i 0.348011i 0.984745 + 0.174005i $$0.0556710\pi$$
−0.984745 + 0.174005i $$0.944329\pi$$
$$318$$ 0 0
$$319$$ 18.3205 1.02575
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 12.0000i 0.668734i
$$323$$ 12.1962i 0.678612i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9.32051 0.516215
$$327$$ 0 0
$$328$$ − 18.2487i − 1.00762i
$$329$$ −41.3205 −2.27807
$$330$$ 0 0
$$331$$ −0.464102 −0.0255093 −0.0127547 0.999919i $$-0.504060\pi$$
−0.0127547 + 0.999919i $$0.504060\pi$$
$$332$$ − 3.21539i − 0.176467i
$$333$$ 0 0
$$334$$ 12.9282 0.707400
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 27.3205i 1.48824i 0.668044 + 0.744121i $$0.267132\pi$$
−0.668044 + 0.744121i $$0.732868\pi$$
$$338$$ 7.94744i 0.432284i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 17.1962 0.931224
$$342$$ 0 0
$$343$$ 39.7128i 2.14429i
$$344$$ 0.497423 0.0268192
$$345$$ 0 0
$$346$$ 6.24871 0.335933
$$347$$ − 28.5885i − 1.53471i −0.641223 0.767354i $$-0.721573\pi$$
0.641223 0.767354i $$-0.278427\pi$$
$$348$$ 0 0
$$349$$ 18.8564 1.00936 0.504680 0.863306i $$-0.331610\pi$$
0.504680 + 0.863306i $$0.331610\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 33.5692i − 1.78925i
$$353$$ − 25.5167i − 1.35811i −0.734085 0.679057i $$-0.762389\pi$$
0.734085 0.679057i $$-0.237611\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −7.60770 −0.403207
$$357$$ 0 0
$$358$$ − 5.94744i − 0.314332i
$$359$$ 6.12436 0.323231 0.161616 0.986854i $$-0.448330\pi$$
0.161616 + 0.986854i $$0.448330\pi$$
$$360$$ 0 0
$$361$$ 0.928203 0.0488528
$$362$$ 19.3731i 1.01823i
$$363$$ 0 0
$$364$$ −10.1436 −0.531669
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 31.1769i 1.62742i 0.581270 + 0.813711i $$0.302556\pi$$
−0.581270 + 0.813711i $$0.697444\pi$$
$$368$$ 3.71281i 0.193544i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −31.8564 −1.65390
$$372$$ 0 0
$$373$$ − 20.0526i − 1.03828i −0.854689 0.519141i $$-0.826252\pi$$
0.854689 0.519141i $$-0.173748\pi$$
$$374$$ −11.4641 −0.592795
$$375$$ 0 0
$$376$$ 22.1436 1.14197
$$377$$ 4.67949i 0.241006i
$$378$$ 0 0
$$379$$ −2.39230 −0.122884 −0.0614422 0.998111i $$-0.519570\pi$$
−0.0614422 + 0.998111i $$0.519570\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 5.94744i − 0.304298i
$$383$$ − 2.53590i − 0.129578i −0.997899 0.0647892i $$-0.979363\pi$$
0.997899 0.0647892i $$-0.0206375\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3.85641 −0.196286
$$387$$ 0 0
$$388$$ − 14.1436i − 0.718032i
$$389$$ −27.4641 −1.39249 −0.696243 0.717807i $$-0.745146\pi$$
−0.696243 + 0.717807i $$0.745146\pi$$
$$390$$ 0 0
$$391$$ 9.46410 0.478620
$$392$$ − 39.0333i − 1.97148i
$$393$$ 0 0
$$394$$ 10.1436 0.511027
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.3923i 0.722329i 0.932502 + 0.361165i $$0.117621\pi$$
−0.932502 + 0.361165i $$0.882379\pi$$
$$398$$ 1.46410i 0.0733888i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.9282 1.24486 0.622428 0.782677i $$-0.286147\pi$$
0.622428 + 0.782677i $$0.286147\pi$$
$$402$$ 0 0
$$403$$ 4.39230i 0.218796i
$$404$$ −3.89488 −0.193778
$$405$$ 0 0
$$406$$ −11.0718 −0.549484
$$407$$ 15.6603i 0.776250i
$$408$$ 0 0
$$409$$ 17.8564 0.882942 0.441471 0.897275i $$-0.354457\pi$$
0.441471 + 0.897275i $$0.354457\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0.784610i 0.0386549i
$$413$$ − 39.1244i − 1.92518i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.57437 0.420393
$$417$$ 0 0
$$418$$ 18.7321i 0.916215i
$$419$$ −20.3923 −0.996229 −0.498115 0.867111i $$-0.665974\pi$$
−0.498115 + 0.867111i $$0.665974\pi$$
$$420$$ 0 0
$$421$$ 33.7846 1.64656 0.823281 0.567635i $$-0.192141\pi$$
0.823281 + 0.567635i $$0.192141\pi$$
$$422$$ − 6.48334i − 0.315604i
$$423$$ 0 0
$$424$$ 17.0718 0.829080
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 18.9282i − 0.916000i
$$428$$ − 12.4974i − 0.604086i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −21.3397 −1.02790 −0.513950 0.857820i $$-0.671818\pi$$
−0.513950 + 0.857820i $$0.671818\pi$$
$$432$$ 0 0
$$433$$ 35.4641i 1.70430i 0.523301 + 0.852148i $$0.324700\pi$$
−0.523301 + 0.852148i $$0.675300\pi$$
$$434$$ −10.3923 −0.498847
$$435$$ 0 0
$$436$$ 8.88973 0.425741
$$437$$ − 15.4641i − 0.739748i
$$438$$ 0 0
$$439$$ −5.39230 −0.257361 −0.128680 0.991686i $$-0.541074\pi$$
−0.128680 + 0.991686i $$0.541074\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 2.92820i − 0.139280i
$$443$$ 0.339746i 0.0161418i 0.999967 + 0.00807091i $$0.00256908\pi$$
−0.999967 + 0.00807091i $$0.997431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 12.2872 0.581815
$$447$$ 0 0
$$448$$ 10.1436i 0.479240i
$$449$$ −8.12436 −0.383412 −0.191706 0.981452i $$-0.561402\pi$$
−0.191706 + 0.981452i $$0.561402\pi$$
$$450$$ 0 0
$$451$$ 41.2487 1.94233
$$452$$ 28.0000i 1.31701i
$$453$$ 0 0
$$454$$ 13.2154 0.620229
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2.73205i 0.127800i 0.997956 + 0.0639000i $$0.0203539\pi$$
−0.997956 + 0.0639000i $$0.979646\pi$$
$$458$$ 8.78461i 0.410478i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −37.0526 −1.72571 −0.862855 0.505452i $$-0.831326\pi$$
−0.862855 + 0.505452i $$0.831326\pi$$
$$462$$ 0 0
$$463$$ 10.3923i 0.482971i 0.970404 + 0.241486i $$0.0776347\pi$$
−0.970404 + 0.241486i $$0.922365\pi$$
$$464$$ −3.42563 −0.159031
$$465$$ 0 0
$$466$$ −20.5359 −0.951307
$$467$$ 37.3731i 1.72942i 0.502272 + 0.864710i $$0.332498\pi$$
−0.502272 + 0.864710i $$0.667502\pi$$
$$468$$ 0 0
$$469$$ 16.3923 0.756926
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 20.9667i 0.965070i
$$473$$ 1.12436i 0.0516979i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −18.9282 −0.867573
$$477$$ 0 0
$$478$$ 0.392305i 0.0179436i
$$479$$ 11.8756 0.542612 0.271306 0.962493i $$-0.412544\pi$$
0.271306 + 0.962493i $$0.412544\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ − 11.9474i − 0.544191i
$$483$$ 0 0
$$484$$ 32.0000 1.45455
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 24.3923i 1.10532i 0.833407 + 0.552660i $$0.186387\pi$$
−0.833407 + 0.552660i $$0.813613\pi$$
$$488$$ 10.1436i 0.459179i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13.8756 −0.626199 −0.313100 0.949720i $$-0.601367\pi$$
−0.313100 + 0.949720i $$0.601367\pi$$
$$492$$ 0 0
$$493$$ 8.73205i 0.393272i
$$494$$ −4.78461 −0.215270
$$495$$ 0 0
$$496$$ −3.21539 −0.144375
$$497$$ 17.6603i 0.792171i
$$498$$ 0 0
$$499$$ −24.3205 −1.08874 −0.544368 0.838847i $$-0.683230\pi$$
−0.544368 + 0.838847i $$0.683230\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 7.60770i − 0.339548i
$$503$$ 7.32051i 0.326405i 0.986593 + 0.163203i $$0.0521824\pi$$
−0.986593 + 0.163203i $$0.947818\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 14.5359 0.646200
$$507$$ 0 0
$$508$$ − 21.3590i − 0.947652i
$$509$$ 6.78461 0.300723 0.150361 0.988631i $$-0.451956\pi$$
0.150361 + 0.988631i $$0.451956\pi$$
$$510$$ 0 0
$$511$$ 36.2487 1.60355
$$512$$ 11.7128i 0.517638i
$$513$$ 0 0
$$514$$ −7.60770 −0.335561
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 50.0526i 2.20131i
$$518$$ − 9.46410i − 0.415829i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 19.4641 0.852738 0.426369 0.904549i $$-0.359793\pi$$
0.426369 + 0.904549i $$0.359793\pi$$
$$522$$ 0 0
$$523$$ − 22.2487i − 0.972868i −0.873717 0.486434i $$-0.838297\pi$$
0.873717 0.486434i $$-0.161703\pi$$
$$524$$ 22.8231 0.997031
$$525$$ 0 0
$$526$$ 15.6077 0.680528
$$527$$ 8.19615i 0.357030i
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 30.9282i 1.34091i
$$533$$ 10.5359i 0.456360i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −8.78461 −0.379437
$$537$$ 0 0
$$538$$ − 7.80385i − 0.336448i
$$539$$ 88.2295 3.80031
$$540$$ 0 0
$$541$$ −24.4641 −1.05179 −0.525897 0.850548i $$-0.676270\pi$$
−0.525897 + 0.850548i $$0.676270\pi$$
$$542$$ − 2.14359i − 0.0920752i
$$543$$ 0 0
$$544$$ 16.0000 0.685994
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 33.8564i − 1.44760i −0.690012 0.723798i $$-0.742395\pi$$
0.690012 0.723798i $$-0.257605\pi$$
$$548$$ 3.71281i 0.158604i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 14.2679 0.607835
$$552$$ 0 0
$$553$$ 73.1769i 3.11180i
$$554$$ −2.78461 −0.118307
$$555$$ 0 0
$$556$$ −0.889727 −0.0377328
$$557$$ − 2.53590i − 0.107449i −0.998556 0.0537247i $$-0.982891\pi$$
0.998556 0.0537247i $$-0.0171094\pi$$
$$558$$ 0 0
$$559$$ −0.287187 −0.0121467
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 11.3205i 0.477527i
$$563$$ 16.7321i 0.705172i 0.935779 + 0.352586i $$0.114698\pi$$
−0.935779 + 0.352586i $$0.885302\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −21.4641 −0.902203
$$567$$ 0 0
$$568$$ − 9.46410i − 0.397105i
$$569$$ 32.9090 1.37962 0.689808 0.723993i $$-0.257695\pi$$
0.689808 + 0.723993i $$0.257695\pi$$
$$570$$ 0 0
$$571$$ −39.7846 −1.66493 −0.832467 0.554075i $$-0.813072\pi$$
−0.832467 + 0.554075i $$0.813072\pi$$
$$572$$ 12.2872i 0.513753i
$$573$$ 0 0
$$574$$ −24.9282 −1.04048
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15.2679i 0.635613i 0.948156 + 0.317807i $$0.102946\pi$$
−0.948156 + 0.317807i $$0.897054\pi$$
$$578$$ 6.98076i 0.290361i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −10.3923 −0.431145
$$582$$ 0 0
$$583$$ 38.5885i 1.59817i
$$584$$ −19.4256 −0.803838
$$585$$ 0 0
$$586$$ 21.0333 0.868878
$$587$$ − 11.6603i − 0.481270i −0.970616 0.240635i $$-0.922644\pi$$
0.970616 0.240635i $$-0.0773557\pi$$
$$588$$ 0 0
$$589$$ 13.3923 0.551820
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 2.92820i − 0.120348i
$$593$$ 0.143594i 0.00589668i 0.999996 + 0.00294834i $$0.000938487\pi$$
−0.999996 + 0.00294834i $$0.999062\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 11.7128 0.479776
$$597$$ 0 0
$$598$$ 3.71281i 0.151828i
$$599$$ −27.1962 −1.11120 −0.555602 0.831448i $$-0.687512\pi$$
−0.555602 + 0.831448i $$0.687512\pi$$
$$600$$ 0 0
$$601$$ −31.2487 −1.27466 −0.637331 0.770590i $$-0.719961\pi$$
−0.637331 + 0.770590i $$0.719961\pi$$
$$602$$ − 0.679492i − 0.0276940i
$$603$$ 0 0
$$604$$ 7.89488 0.321238
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 16.1962i − 0.657382i −0.944438 0.328691i $$-0.893393\pi$$
0.944438 0.328691i $$-0.106607\pi$$
$$608$$ − 26.1436i − 1.06026i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12.7846 −0.517210
$$612$$ 0 0
$$613$$ − 1.46410i − 0.0591345i −0.999563 0.0295673i $$-0.990587\pi$$
0.999563 0.0295673i $$-0.00941292\pi$$
$$614$$ −10.2872 −0.415157
$$615$$ 0 0
$$616$$ −68.7846 −2.77141
$$617$$ − 6.92820i − 0.278919i −0.990228 0.139459i $$-0.955464\pi$$
0.990228 0.139459i $$-0.0445365\pi$$
$$618$$ 0 0
$$619$$ 11.8564 0.476549 0.238275 0.971198i $$-0.423418\pi$$
0.238275 + 0.971198i $$0.423418\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 14.4449i 0.579186i
$$623$$ 24.5885i 0.985116i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −6.64102 −0.265428
$$627$$ 0 0
$$628$$ − 28.0000i − 1.11732i
$$629$$ −7.46410 −0.297613
$$630$$ 0 0
$$631$$ −32.7128 −1.30228 −0.651138 0.758959i $$-0.725708\pi$$
−0.651138 + 0.758959i $$0.725708\pi$$
$$632$$ − 39.2154i − 1.55990i
$$633$$ 0 0
$$634$$ −4.53590 −0.180144
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 22.5359i 0.892905i
$$638$$ 13.4115i 0.530968i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9.33975 0.368898 0.184449 0.982842i $$-0.440950\pi$$
0.184449 + 0.982842i $$0.440950\pi$$
$$642$$ 0 0
$$643$$ − 41.6603i − 1.64292i −0.570266 0.821460i $$-0.693160\pi$$
0.570266 0.821460i $$-0.306840\pi$$
$$644$$ 24.0000 0.945732
$$645$$ 0 0
$$646$$ −8.92820 −0.351275
$$647$$ 11.4641i 0.450700i 0.974278 + 0.225350i $$0.0723526\pi$$
−0.974278 + 0.225350i $$0.927647\pi$$
$$648$$ 0 0
$$649$$ −47.3923 −1.86031
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 18.6410i − 0.730039i
$$653$$ 17.4641i 0.683423i 0.939805 + 0.341712i $$0.111007\pi$$
−0.939805 + 0.341712i $$0.888993\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −7.71281 −0.301135
$$657$$ 0 0
$$658$$ − 30.2487i − 1.17922i
$$659$$ 1.46410 0.0570333 0.0285167 0.999593i $$-0.490922\pi$$
0.0285167 + 0.999593i $$0.490922\pi$$
$$660$$ 0 0
$$661$$ 18.3205 0.712585 0.356293 0.934374i $$-0.384041\pi$$
0.356293 + 0.934374i $$0.384041\pi$$
$$662$$ − 0.339746i − 0.0132046i
$$663$$ 0 0
$$664$$ 5.56922 0.216128
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 11.0718i − 0.428702i
$$668$$ − 25.8564i − 1.00041i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −22.9282 −0.885133
$$672$$ 0 0
$$673$$ 10.3923i 0.400594i 0.979735 + 0.200297i $$0.0641907\pi$$
−0.979735 + 0.200297i $$0.935809\pi$$
$$674$$ −20.0000 −0.770371
$$675$$ 0 0
$$676$$ 15.8949 0.611342
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ −45.7128 −1.75430
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.5885i 0.482037i
$$683$$ 19.6077i 0.750268i 0.926971 + 0.375134i $$0.122403\pi$$
−0.926971 + 0.375134i $$0.877597\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −29.0718 −1.10997
$$687$$ 0 0
$$688$$ − 0.210236i − 0.00801515i
$$689$$ −9.85641 −0.375499
$$690$$ 0 0
$$691$$ −17.7128 −0.673827 −0.336914 0.941536i $$-0.609383\pi$$
−0.336914 + 0.941536i $$0.609383\pi$$
$$692$$ − 12.4974i − 0.475081i
$$693$$ 0 0
$$694$$ 20.9282 0.794424
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 19.6603i 0.744685i
$$698$$ 13.8038i 0.522483i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −20.8038 −0.785750 −0.392875 0.919592i $$-0.628520\pi$$
−0.392875 + 0.919592i $$0.628520\pi$$
$$702$$ 0 0
$$703$$ 12.1962i 0.459987i
$$704$$ 12.2872 0.463091
$$705$$ 0 0
$$706$$ 18.6795 0.703012
$$707$$ 12.5885i 0.473438i
$$708$$ 0 0
$$709$$ 22.5359 0.846353 0.423177 0.906047i $$-0.360915\pi$$
0.423177 + 0.906047i $$0.360915\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 13.1769i − 0.493826i
$$713$$ − 10.3923i − 0.389195i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −11.8949 −0.444533
$$717$$ 0 0
$$718$$ 4.48334i 0.167317i
$$719$$ −8.41154 −0.313698 −0.156849 0.987623i $$-0.550134\pi$$
−0.156849 + 0.987623i $$0.550134\pi$$
$$720$$ 0 0
$$721$$ 2.53590 0.0944418
$$722$$ 0.679492i 0.0252881i
$$723$$ 0 0
$$724$$ 38.7461 1.43999
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 8.39230i − 0.311253i −0.987816 0.155627i $$-0.950260\pi$$
0.987816 0.155627i $$-0.0497397\pi$$
$$728$$ − 17.5692i − 0.651159i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −0.535898 −0.0198209
$$732$$ 0 0
$$733$$ − 34.7846i − 1.28480i −0.766370 0.642399i $$-0.777939\pi$$
0.766370 0.642399i $$-0.222061\pi$$
$$734$$ −22.8231 −0.842415
$$735$$ 0 0
$$736$$ −20.2872 −0.747796
$$737$$ − 19.8564i − 0.731420i
$$738$$ 0 0
$$739$$ −22.4641 −0.826355 −0.413178 0.910650i $$-0.635581\pi$$
−0.413178 + 0.910650i $$0.635581\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 23.3205i − 0.856123i
$$743$$ − 19.9090i − 0.730389i −0.930931 0.365195i $$-0.881002\pi$$
0.930931 0.365195i $$-0.118998\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 14.6795 0.537454
$$747$$ 0 0
$$748$$ 22.9282i 0.838338i
$$749$$ −40.3923 −1.47590
$$750$$ 0 0
$$751$$ 48.7846 1.78018 0.890088 0.455789i $$-0.150643\pi$$
0.890088 + 0.455789i $$0.150643\pi$$
$$752$$ − 9.35898i − 0.341287i
$$753$$ 0 0
$$754$$ −3.42563 −0.124754
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 9.17691i − 0.333541i −0.985996 0.166770i $$-0.946666\pi$$
0.985996 0.166770i $$-0.0533338\pi$$
$$758$$ − 1.75129i − 0.0636097i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 23.4449 0.849876 0.424938 0.905223i $$-0.360296\pi$$
0.424938 + 0.905223i $$0.360296\pi$$
$$762$$ 0 0
$$763$$ − 28.7321i − 1.04017i
$$764$$ −11.8949 −0.430342
$$765$$ 0 0
$$766$$ 1.85641 0.0670747
$$767$$ − 12.1051i − 0.437090i
$$768$$ 0 0
$$769$$ 9.53590 0.343873 0.171937 0.985108i $$-0.444998\pi$$
0.171937 + 0.985108i $$0.444998\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 7.71281i 0.277590i
$$773$$ − 1.51666i − 0.0545505i −0.999628 0.0272752i $$-0.991317\pi$$
0.999628 0.0272752i $$-0.00868306\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 24.4974 0.879406
$$777$$ 0 0
$$778$$ − 20.1051i − 0.720803i
$$779$$ 32.1244 1.15097
$$780$$ 0 0
$$781$$ 21.3923 0.765477
$$782$$ 6.92820i 0.247752i
$$783$$ 0 0
$$784$$ −16.4974 −0.589194
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 48.0526i − 1.71289i −0.516239 0.856444i $$-0.672668\pi$$
0.516239 0.856444i $$-0.327332\pi$$
$$788$$ − 20.2872i − 0.722701i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 90.4974 3.21772
$$792$$ 0 0
$$793$$ − 5.85641i − 0.207967i
$$794$$ −10.5359 −0.373905
$$795$$ 0 0
$$796$$ 2.92820 0.103787
$$797$$ − 15.6077i − 0.552853i −0.961035 0.276426i $$-0.910850\pi$$
0.961035 0.276426i $$-0.0891502\pi$$
$$798$$ 0 0
$$799$$ −23.8564 −0.843979
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 18.2487i 0.644384i
$$803$$ − 43.9090i − 1.54951i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3.21539 −0.113257
$$807$$ 0 0
$$808$$ − 6.74613i − 0.237328i
$$809$$ −45.4449 −1.59776 −0.798878 0.601493i $$-0.794573\pi$$
−0.798878 + 0.601493i $$0.794573\pi$$
$$810$$ 0 0
$$811$$ −18.4641 −0.648362 −0.324181 0.945995i $$-0.605089\pi$$
−0.324181 + 0.945995i $$0.605089\pi$$
$$812$$ 22.1436i 0.777088i
$$813$$ 0 0
$$814$$ −11.4641 −0.401817
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0.875644i 0.0306349i
$$818$$ 13.0718i 0.457045i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −37.7321 −1.31686 −0.658429 0.752643i $$-0.728779\pi$$
−0.658429 + 0.752643i $$0.728779\pi$$
$$822$$ 0 0
$$823$$ − 3.85641i − 0.134426i −0.997739 0.0672129i $$-0.978589\pi$$
0.997739 0.0672129i $$-0.0214107\pi$$
$$824$$ −1.35898 −0.0473424
$$825$$ 0 0
$$826$$ 28.6410 0.996548
$$827$$ − 11.6077i − 0.403639i −0.979423 0.201820i $$-0.935315\pi$$
0.979423 0.201820i $$-0.0646855\pi$$
$$828$$ 0 0
$$829$$ 23.7846 0.826074 0.413037 0.910714i $$-0.364468\pi$$
0.413037 + 0.910714i $$0.364468\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 3.13844i 0.108806i
$$833$$ 42.0526i 1.45703i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 37.4641 1.29572
$$837$$ 0 0
$$838$$ − 14.9282i − 0.515686i
$$839$$ −33.1962 −1.14606 −0.573029 0.819535i $$-0.694232\pi$$
−0.573029 + 0.819535i $$0.694232\pi$$
$$840$$ 0 0
$$841$$ −18.7846 −0.647745
$$842$$ 24.7321i 0.852323i
$$843$$ 0 0
$$844$$ −12.9667 −0.446331
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 103.426i − 3.55375i
$$848$$ − 7.21539i − 0.247778i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 9.46410 0.324425
$$852$$ 0 0
$$853$$ − 10.4833i − 0.358943i −0.983763 0.179471i $$-0.942561\pi$$
0.983763 0.179471i $$-0.0574387\pi$$
$$854$$ 13.8564 0.474156
$$855$$ 0 0
$$856$$ 21.6462 0.739851
$$857$$ 44.4974i 1.52000i 0.649921 + 0.760002i $$0.274802\pi$$
−0.649921 + 0.760002i $$0.725198\pi$$
$$858$$ 0 0
$$859$$ 11.0000 0.375315 0.187658 0.982235i $$-0.439910\pi$$
0.187658 + 0.982235i $$0.439910\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 15.6218i − 0.532080i
$$863$$ 23.1244i 0.787162i 0.919290 + 0.393581i $$0.128764\pi$$
−0.919290 + 0.393581i $$0.871236\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −25.9615 −0.882209
$$867$$ 0 0
$$868$$ 20.7846i 0.705476i
$$869$$ 88.6410 3.00694
$$870$$ 0 0
$$871$$ 5.07180 0.171851
$$872$$ 15.3975i 0.521424i
$$873$$ 0 0
$$874$$ 11.3205 0.382922
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 33.4641i 1.13000i 0.825090 + 0.565001i $$0.191124\pi$$
−0.825090 + 0.565001i $$0.808876\pi$$
$$878$$ − 3.94744i − 0.133220i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −47.0526 −1.58524 −0.792620 0.609715i $$-0.791284\pi$$
−0.792620 + 0.609715i $$0.791284\pi$$
$$882$$ 0 0
$$883$$ 30.1962i 1.01618i 0.861304 + 0.508091i $$0.169649\pi$$
−0.861304 + 0.508091i $$0.830351\pi$$
$$884$$ −5.85641 −0.196972
$$885$$ 0 0
$$886$$ −0.248711 −0.00835562
$$887$$ 33.8038i 1.13502i 0.823366 + 0.567511i $$0.192094\pi$$
−0.823366 + 0.567511i $$0.807906\pi$$
$$888$$ 0 0
$$889$$ −69.0333 −2.31530
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 24.5744i − 0.822811i
$$893$$ 38.9808i 1.30444i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 48.0000 1.60357
$$897$$ 0 0
$$898$$ − 5.94744i − 0.198469i
$$899$$ 9.58846 0.319793
$$900$$ 0 0
$$901$$ −18.3923 −0.612737
$$902$$ 30.1962i 1.00542i
$$903$$ 0 0
$$904$$ −48.4974 −1.61300
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 30.0000i − 0.996134i −0.867139 0.498067i $$-0.834043\pi$$
0.867139 0.498067i $$-0.165957\pi$$
$$908$$ − 26.4308i − 0.877136i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 23.5885 0.781520 0.390760 0.920493i $$-0.372212\pi$$
0.390760 + 0.920493i $$0.372212\pi$$
$$912$$ 0 0
$$913$$ 12.5885i 0.416617i
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 17.5692 0.580503
$$917$$ − 73.7654i − 2.43595i
$$918$$ 0 0
$$919$$ 56.9615 1.87899 0.939494 0.342566i $$-0.111296\pi$$
0.939494 + 0.342566i $$0.111296\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 27.1244i − 0.893293i
$$923$$ 5.46410i 0.179853i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −7.60770 −0.250004
$$927$$ 0 0
$$928$$ − 18.7180i − 0.614447i
$$929$$ 44.3731 1.45583 0.727917 0.685666i $$-0.240489\pi$$
0.727917 + 0.685666i $$0.240489\pi$$
$$930$$ 0 0
$$931$$ 68.7128 2.25197
$$932$$ 41.0718i 1.34535i
$$933$$ 0 0
$$934$$ −27.3590 −0.895213
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 4.14359i − 0.135365i −0.997707 0.0676827i $$-0.978439\pi$$
0.997707 0.0676827i $$-0.0215605\pi$$
$$938$$ 12.0000i 0.391814i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 35.1769 1.14673 0.573367 0.819298i $$-0.305637\pi$$
0.573367 + 0.819298i $$0.305637\pi$$
$$942$$ 0 0
$$943$$ − 24.9282i − 0.811774i
$$944$$ 8.86156 0.288419
$$945$$ 0 0
$$946$$ −0.823085 −0.0267608
$$947$$ − 57.7128i − 1.87541i −0.347427 0.937707i $$-0.612944\pi$$
0.347427 0.937707i $$-0.387056\pi$$
$$948$$ 0 0
$$949$$ 11.2154 0.364067
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 32.7846i − 1.06256i
$$953$$ − 36.3923i − 1.17886i −0.807819 0.589431i $$-0.799352\pi$$
0.807819 0.589431i $$-0.200648\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0.784610 0.0253761
$$957$$ 0 0
$$958$$ 8.69358i 0.280877i
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ − 2.92820i − 0.0944091i
$$963$$ 0 0
$$964$$ −23.8949 −0.769602
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 25.8038i − 0.829796i −0.909868 0.414898i $$-0.863817\pi$$
0.909868 0.414898i $$-0.136183\pi$$
$$968$$ 55.4256i 1.78145i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 17.4449 0.559832 0.279916 0.960024i $$-0.409693\pi$$
0.279916 + 0.960024i $$0.409693\pi$$
$$972$$ 0 0
$$973$$ 2.87564i 0.0921889i
$$974$$ −17.8564 −0.572156
$$975$$ 0 0
$$976$$ 4.28719 0.137230
$$977$$ 5.46410i 0.174812i 0.996173 + 0.0874060i $$0.0278578\pi$$
−0.996173 + 0.0874060i $$0.972142\pi$$
$$978$$ 0 0
$$979$$ 29.7846 0.951920
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 10.1577i − 0.324144i
$$983$$ − 48.5885i − 1.54973i −0.632126 0.774866i $$-0.717818\pi$$
0.632126 0.774866i $$-0.282182\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −6.39230 −0.203572
$$987$$ 0 0
$$988$$ 9.56922i 0.304437i
$$989$$ 0.679492 0.0216066
$$990$$ 0 0
$$991$$ 30.8564 0.980186 0.490093 0.871670i $$-0.336963\pi$$
0.490093 + 0.871670i $$0.336963\pi$$
$$992$$ − 17.5692i − 0.557823i
$$993$$ 0 0
$$994$$ −12.9282 −0.410058
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 25.5167i 0.808121i 0.914732 + 0.404060i $$0.132401\pi$$
−0.914732 + 0.404060i $$0.867599\pi$$
$$998$$ − 17.8038i − 0.563571i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2025.2.b.g.649.3 4
3.2 odd 2 2025.2.b.h.649.2 4
5.2 odd 4 2025.2.a.m.1.1 2
5.3 odd 4 405.2.a.g.1.2 2
5.4 even 2 inner 2025.2.b.g.649.2 4
15.2 even 4 2025.2.a.g.1.2 2
15.8 even 4 405.2.a.h.1.1 yes 2
15.14 odd 2 2025.2.b.h.649.3 4
20.3 even 4 6480.2.a.br.1.2 2
45.13 odd 12 405.2.e.l.136.1 4
45.23 even 12 405.2.e.i.136.2 4
45.38 even 12 405.2.e.i.271.2 4
45.43 odd 12 405.2.e.l.271.1 4
60.23 odd 4 6480.2.a.bi.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.2 2 5.3 odd 4
405.2.a.h.1.1 yes 2 15.8 even 4
405.2.e.i.136.2 4 45.23 even 12
405.2.e.i.271.2 4 45.38 even 12
405.2.e.l.136.1 4 45.13 odd 12
405.2.e.l.271.1 4 45.43 odd 12
2025.2.a.g.1.2 2 15.2 even 4
2025.2.a.m.1.1 2 5.2 odd 4
2025.2.b.g.649.2 4 5.4 even 2 inner
2025.2.b.g.649.3 4 1.1 even 1 trivial
2025.2.b.h.649.2 4 3.2 odd 2
2025.2.b.h.649.3 4 15.14 odd 2
6480.2.a.bi.1.2 2 60.23 odd 4
6480.2.a.br.1.2 2 20.3 even 4